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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    <a name="line.17"></a>
<FONT color="green">018</FONT>    package org.apache.commons.math3.ode;<a name="line.18"></a>
<FONT color="green">019</FONT>    <a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.DimensionMismatchException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.MaxCountExceededException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.NoBracketingException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.NumberIsTooSmallException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.linear.Array2DRowRealMatrix;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator;<a name="line.26"></a>
<FONT color="green">027</FONT>    import org.apache.commons.math3.ode.nonstiff.DormandPrince853Integrator;<a name="line.27"></a>
<FONT color="green">028</FONT>    import org.apache.commons.math3.ode.sampling.StepHandler;<a name="line.28"></a>
<FONT color="green">029</FONT>    import org.apache.commons.math3.ode.sampling.StepInterpolator;<a name="line.29"></a>
<FONT color="green">030</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.30"></a>
<FONT color="green">031</FONT>    <a name="line.31"></a>
<FONT color="green">032</FONT>    /**<a name="line.32"></a>
<FONT color="green">033</FONT>     * This class is the base class for multistep integrators for Ordinary<a name="line.33"></a>
<FONT color="green">034</FONT>     * Differential Equations.<a name="line.34"></a>
<FONT color="green">035</FONT>     * &lt;p&gt;We define scaled derivatives s&lt;sub&gt;i&lt;/sub&gt;(n) at step n as:<a name="line.35"></a>
<FONT color="green">036</FONT>     * &lt;pre&gt;<a name="line.36"></a>
<FONT color="green">037</FONT>     * s&lt;sub&gt;1&lt;/sub&gt;(n) = h y'&lt;sub&gt;n&lt;/sub&gt; for first derivative<a name="line.37"></a>
<FONT color="green">038</FONT>     * s&lt;sub&gt;2&lt;/sub&gt;(n) = h&lt;sup&gt;2&lt;/sup&gt;/2 y''&lt;sub&gt;n&lt;/sub&gt; for second derivative<a name="line.38"></a>
<FONT color="green">039</FONT>     * s&lt;sub&gt;3&lt;/sub&gt;(n) = h&lt;sup&gt;3&lt;/sup&gt;/6 y'''&lt;sub&gt;n&lt;/sub&gt; for third derivative<a name="line.39"></a>
<FONT color="green">040</FONT>     * ...<a name="line.40"></a>
<FONT color="green">041</FONT>     * s&lt;sub&gt;k&lt;/sub&gt;(n) = h&lt;sup&gt;k&lt;/sup&gt;/k! y&lt;sup&gt;(k)&lt;/sup&gt;&lt;sub&gt;n&lt;/sub&gt; for k&lt;sup&gt;th&lt;/sup&gt; derivative<a name="line.41"></a>
<FONT color="green">042</FONT>     * &lt;/pre&gt;&lt;/p&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     * &lt;p&gt;Rather than storing several previous steps separately, this implementation uses<a name="line.43"></a>
<FONT color="green">044</FONT>     * the Nordsieck vector with higher degrees scaled derivatives all taken at the same<a name="line.44"></a>
<FONT color="green">045</FONT>     * step (y&lt;sub&gt;n&lt;/sub&gt;, s&lt;sub&gt;1&lt;/sub&gt;(n) and r&lt;sub&gt;n&lt;/sub&gt;) where r&lt;sub&gt;n&lt;/sub&gt; is defined as:<a name="line.45"></a>
<FONT color="green">046</FONT>     * &lt;pre&gt;<a name="line.46"></a>
<FONT color="green">047</FONT>     * r&lt;sub&gt;n&lt;/sub&gt; = [ s&lt;sub&gt;2&lt;/sub&gt;(n), s&lt;sub&gt;3&lt;/sub&gt;(n) ... s&lt;sub&gt;k&lt;/sub&gt;(n) ]&lt;sup&gt;T&lt;/sup&gt;<a name="line.47"></a>
<FONT color="green">048</FONT>     * &lt;/pre&gt;<a name="line.48"></a>
<FONT color="green">049</FONT>     * (we omit the k index in the notation for clarity)&lt;/p&gt;<a name="line.49"></a>
<FONT color="green">050</FONT>     * &lt;p&gt;<a name="line.50"></a>
<FONT color="green">051</FONT>     * Multistep integrators with Nordsieck representation are highly sensitive to<a name="line.51"></a>
<FONT color="green">052</FONT>     * large step changes because when the step is multiplied by factor a, the<a name="line.52"></a>
<FONT color="green">053</FONT>     * k&lt;sup&gt;th&lt;/sup&gt; component of the Nordsieck vector is multiplied by a&lt;sup&gt;k&lt;/sup&gt;<a name="line.53"></a>
<FONT color="green">054</FONT>     * and the last components are the least accurate ones. The default max growth<a name="line.54"></a>
<FONT color="green">055</FONT>     * factor is therefore set to a quite low value: 2&lt;sup&gt;1/order&lt;/sup&gt;.<a name="line.55"></a>
<FONT color="green">056</FONT>     * &lt;/p&gt;<a name="line.56"></a>
<FONT color="green">057</FONT>     *<a name="line.57"></a>
<FONT color="green">058</FONT>     * @see org.apache.commons.math3.ode.nonstiff.AdamsBashforthIntegrator<a name="line.58"></a>
<FONT color="green">059</FONT>     * @see org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator<a name="line.59"></a>
<FONT color="green">060</FONT>     * @version $Id: MultistepIntegrator.java 1421448 2012-12-13 19:45:57Z tn $<a name="line.60"></a>
<FONT color="green">061</FONT>     * @since 2.0<a name="line.61"></a>
<FONT color="green">062</FONT>     */<a name="line.62"></a>
<FONT color="green">063</FONT>    public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator {<a name="line.63"></a>
<FONT color="green">064</FONT>    <a name="line.64"></a>
<FONT color="green">065</FONT>        /** First scaled derivative (h y'). */<a name="line.65"></a>
<FONT color="green">066</FONT>        protected double[] scaled;<a name="line.66"></a>
<FONT color="green">067</FONT>    <a name="line.67"></a>
<FONT color="green">068</FONT>        /** Nordsieck matrix of the higher scaled derivatives.<a name="line.68"></a>
<FONT color="green">069</FONT>         * &lt;p&gt;(h&lt;sup&gt;2&lt;/sup&gt;/2 y'', h&lt;sup&gt;3&lt;/sup&gt;/6 y''' ..., h&lt;sup&gt;k&lt;/sup&gt;/k! y&lt;sup&gt;(k)&lt;/sup&gt;)&lt;/p&gt;<a name="line.69"></a>
<FONT color="green">070</FONT>         */<a name="line.70"></a>
<FONT color="green">071</FONT>        protected Array2DRowRealMatrix nordsieck;<a name="line.71"></a>
<FONT color="green">072</FONT>    <a name="line.72"></a>
<FONT color="green">073</FONT>        /** Starter integrator. */<a name="line.73"></a>
<FONT color="green">074</FONT>        private FirstOrderIntegrator starter;<a name="line.74"></a>
<FONT color="green">075</FONT>    <a name="line.75"></a>
<FONT color="green">076</FONT>        /** Number of steps of the multistep method (excluding the one being computed). */<a name="line.76"></a>
<FONT color="green">077</FONT>        private final int nSteps;<a name="line.77"></a>
<FONT color="green">078</FONT>    <a name="line.78"></a>
<FONT color="green">079</FONT>        /** Stepsize control exponent. */<a name="line.79"></a>
<FONT color="green">080</FONT>        private double exp;<a name="line.80"></a>
<FONT color="green">081</FONT>    <a name="line.81"></a>
<FONT color="green">082</FONT>        /** Safety factor for stepsize control. */<a name="line.82"></a>
<FONT color="green">083</FONT>        private double safety;<a name="line.83"></a>
<FONT color="green">084</FONT>    <a name="line.84"></a>
<FONT color="green">085</FONT>        /** Minimal reduction factor for stepsize control. */<a name="line.85"></a>
<FONT color="green">086</FONT>        private double minReduction;<a name="line.86"></a>
<FONT color="green">087</FONT>    <a name="line.87"></a>
<FONT color="green">088</FONT>        /** Maximal growth factor for stepsize control. */<a name="line.88"></a>
<FONT color="green">089</FONT>        private double maxGrowth;<a name="line.89"></a>
<FONT color="green">090</FONT>    <a name="line.90"></a>
<FONT color="green">091</FONT>        /**<a name="line.91"></a>
<FONT color="green">092</FONT>         * Build a multistep integrator with the given stepsize bounds.<a name="line.92"></a>
<FONT color="green">093</FONT>         * &lt;p&gt;The default starter integrator is set to the {@link<a name="line.93"></a>
<FONT color="green">094</FONT>         * DormandPrince853Integrator Dormand-Prince 8(5,3)} integrator with<a name="line.94"></a>
<FONT color="green">095</FONT>         * some defaults settings.&lt;/p&gt;<a name="line.95"></a>
<FONT color="green">096</FONT>         * &lt;p&gt;<a name="line.96"></a>
<FONT color="green">097</FONT>         * The default max growth factor is set to a quite low value: 2&lt;sup&gt;1/order&lt;/sup&gt;.<a name="line.97"></a>
<FONT color="green">098</FONT>         * &lt;/p&gt;<a name="line.98"></a>
<FONT color="green">099</FONT>         * @param name name of the method<a name="line.99"></a>
<FONT color="green">100</FONT>         * @param nSteps number of steps of the multistep method<a name="line.100"></a>
<FONT color="green">101</FONT>         * (excluding the one being computed)<a name="line.101"></a>
<FONT color="green">102</FONT>         * @param order order of the method<a name="line.102"></a>
<FONT color="green">103</FONT>         * @param minStep minimal step (must be positive even for backward<a name="line.103"></a>
<FONT color="green">104</FONT>         * integration), the last step can be smaller than this<a name="line.104"></a>
<FONT color="green">105</FONT>         * @param maxStep maximal step (must be positive even for backward<a name="line.105"></a>
<FONT color="green">106</FONT>         * integration)<a name="line.106"></a>
<FONT color="green">107</FONT>         * @param scalAbsoluteTolerance allowed absolute error<a name="line.107"></a>
<FONT color="green">108</FONT>         * @param scalRelativeTolerance allowed relative error<a name="line.108"></a>
<FONT color="green">109</FONT>         * @exception NumberIsTooSmallException if number of steps is smaller than 2<a name="line.109"></a>
<FONT color="green">110</FONT>         */<a name="line.110"></a>
<FONT color="green">111</FONT>        protected MultistepIntegrator(final String name, final int nSteps,<a name="line.111"></a>
<FONT color="green">112</FONT>                                      final int order,<a name="line.112"></a>
<FONT color="green">113</FONT>                                      final double minStep, final double maxStep,<a name="line.113"></a>
<FONT color="green">114</FONT>                                      final double scalAbsoluteTolerance,<a name="line.114"></a>
<FONT color="green">115</FONT>                                      final double scalRelativeTolerance)<a name="line.115"></a>
<FONT color="green">116</FONT>            throws NumberIsTooSmallException {<a name="line.116"></a>
<FONT color="green">117</FONT>    <a name="line.117"></a>
<FONT color="green">118</FONT>            super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);<a name="line.118"></a>
<FONT color="green">119</FONT>    <a name="line.119"></a>
<FONT color="green">120</FONT>            if (nSteps &lt; 2) {<a name="line.120"></a>
<FONT color="green">121</FONT>                throw new NumberIsTooSmallException(<a name="line.121"></a>
<FONT color="green">122</FONT>                      LocalizedFormats.INTEGRATION_METHOD_NEEDS_AT_LEAST_TWO_PREVIOUS_POINTS,<a name="line.122"></a>
<FONT color="green">123</FONT>                      nSteps, 2, true);<a name="line.123"></a>
<FONT color="green">124</FONT>            }<a name="line.124"></a>
<FONT color="green">125</FONT>    <a name="line.125"></a>
<FONT color="green">126</FONT>            starter = new DormandPrince853Integrator(minStep, maxStep,<a name="line.126"></a>
<FONT color="green">127</FONT>                                                     scalAbsoluteTolerance,<a name="line.127"></a>
<FONT color="green">128</FONT>                                                     scalRelativeTolerance);<a name="line.128"></a>
<FONT color="green">129</FONT>            this.nSteps = nSteps;<a name="line.129"></a>
<FONT color="green">130</FONT>    <a name="line.130"></a>
<FONT color="green">131</FONT>            exp = -1.0 / order;<a name="line.131"></a>
<FONT color="green">132</FONT>    <a name="line.132"></a>
<FONT color="green">133</FONT>            // set the default values of the algorithm control parameters<a name="line.133"></a>
<FONT color="green">134</FONT>            setSafety(0.9);<a name="line.134"></a>
<FONT color="green">135</FONT>            setMinReduction(0.2);<a name="line.135"></a>
<FONT color="green">136</FONT>            setMaxGrowth(FastMath.pow(2.0, -exp));<a name="line.136"></a>
<FONT color="green">137</FONT>    <a name="line.137"></a>
<FONT color="green">138</FONT>        }<a name="line.138"></a>
<FONT color="green">139</FONT>    <a name="line.139"></a>
<FONT color="green">140</FONT>        /**<a name="line.140"></a>
<FONT color="green">141</FONT>         * Build a multistep integrator with the given stepsize bounds.<a name="line.141"></a>
<FONT color="green">142</FONT>         * &lt;p&gt;The default starter integrator is set to the {@link<a name="line.142"></a>
<FONT color="green">143</FONT>         * DormandPrince853Integrator Dormand-Prince 8(5,3)} integrator with<a name="line.143"></a>
<FONT color="green">144</FONT>         * some defaults settings.&lt;/p&gt;<a name="line.144"></a>
<FONT color="green">145</FONT>         * &lt;p&gt;<a name="line.145"></a>
<FONT color="green">146</FONT>         * The default max growth factor is set to a quite low value: 2&lt;sup&gt;1/order&lt;/sup&gt;.<a name="line.146"></a>
<FONT color="green">147</FONT>         * &lt;/p&gt;<a name="line.147"></a>
<FONT color="green">148</FONT>         * @param name name of the method<a name="line.148"></a>
<FONT color="green">149</FONT>         * @param nSteps number of steps of the multistep method<a name="line.149"></a>
<FONT color="green">150</FONT>         * (excluding the one being computed)<a name="line.150"></a>
<FONT color="green">151</FONT>         * @param order order of the method<a name="line.151"></a>
<FONT color="green">152</FONT>         * @param minStep minimal step (must be positive even for backward<a name="line.152"></a>
<FONT color="green">153</FONT>         * integration), the last step can be smaller than this<a name="line.153"></a>
<FONT color="green">154</FONT>         * @param maxStep maximal step (must be positive even for backward<a name="line.154"></a>
<FONT color="green">155</FONT>         * integration)<a name="line.155"></a>
<FONT color="green">156</FONT>         * @param vecAbsoluteTolerance allowed absolute error<a name="line.156"></a>
<FONT color="green">157</FONT>         * @param vecRelativeTolerance allowed relative error<a name="line.157"></a>
<FONT color="green">158</FONT>         */<a name="line.158"></a>
<FONT color="green">159</FONT>        protected MultistepIntegrator(final String name, final int nSteps,<a name="line.159"></a>
<FONT color="green">160</FONT>                                      final int order,<a name="line.160"></a>
<FONT color="green">161</FONT>                                      final double minStep, final double maxStep,<a name="line.161"></a>
<FONT color="green">162</FONT>                                      final double[] vecAbsoluteTolerance,<a name="line.162"></a>
<FONT color="green">163</FONT>                                      final double[] vecRelativeTolerance) {<a name="line.163"></a>
<FONT color="green">164</FONT>            super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);<a name="line.164"></a>
<FONT color="green">165</FONT>            starter = new DormandPrince853Integrator(minStep, maxStep,<a name="line.165"></a>
<FONT color="green">166</FONT>                                                     vecAbsoluteTolerance,<a name="line.166"></a>
<FONT color="green">167</FONT>                                                     vecRelativeTolerance);<a name="line.167"></a>
<FONT color="green">168</FONT>            this.nSteps = nSteps;<a name="line.168"></a>
<FONT color="green">169</FONT>    <a name="line.169"></a>
<FONT color="green">170</FONT>            exp = -1.0 / order;<a name="line.170"></a>
<FONT color="green">171</FONT>    <a name="line.171"></a>
<FONT color="green">172</FONT>            // set the default values of the algorithm control parameters<a name="line.172"></a>
<FONT color="green">173</FONT>            setSafety(0.9);<a name="line.173"></a>
<FONT color="green">174</FONT>            setMinReduction(0.2);<a name="line.174"></a>
<FONT color="green">175</FONT>            setMaxGrowth(FastMath.pow(2.0, -exp));<a name="line.175"></a>
<FONT color="green">176</FONT>    <a name="line.176"></a>
<FONT color="green">177</FONT>        }<a name="line.177"></a>
<FONT color="green">178</FONT>    <a name="line.178"></a>
<FONT color="green">179</FONT>        /**<a name="line.179"></a>
<FONT color="green">180</FONT>         * Get the starter integrator.<a name="line.180"></a>
<FONT color="green">181</FONT>         * @return starter integrator<a name="line.181"></a>
<FONT color="green">182</FONT>         */<a name="line.182"></a>
<FONT color="green">183</FONT>        public ODEIntegrator getStarterIntegrator() {<a name="line.183"></a>
<FONT color="green">184</FONT>            return starter;<a name="line.184"></a>
<FONT color="green">185</FONT>        }<a name="line.185"></a>
<FONT color="green">186</FONT>    <a name="line.186"></a>
<FONT color="green">187</FONT>        /**<a name="line.187"></a>
<FONT color="green">188</FONT>         * Set the starter integrator.<a name="line.188"></a>
<FONT color="green">189</FONT>         * &lt;p&gt;The various step and event handlers for this starter integrator<a name="line.189"></a>
<FONT color="green">190</FONT>         * will be managed automatically by the multi-step integrator. Any<a name="line.190"></a>
<FONT color="green">191</FONT>         * user configuration for these elements will be cleared before use.&lt;/p&gt;<a name="line.191"></a>
<FONT color="green">192</FONT>         * @param starterIntegrator starter integrator<a name="line.192"></a>
<FONT color="green">193</FONT>         */<a name="line.193"></a>
<FONT color="green">194</FONT>        public void setStarterIntegrator(FirstOrderIntegrator starterIntegrator) {<a name="line.194"></a>
<FONT color="green">195</FONT>            this.starter = starterIntegrator;<a name="line.195"></a>
<FONT color="green">196</FONT>        }<a name="line.196"></a>
<FONT color="green">197</FONT>    <a name="line.197"></a>
<FONT color="green">198</FONT>        /** Start the integration.<a name="line.198"></a>
<FONT color="green">199</FONT>         * &lt;p&gt;This method computes one step using the underlying starter integrator,<a name="line.199"></a>
<FONT color="green">200</FONT>         * and initializes the Nordsieck vector at step start. The starter integrator<a name="line.200"></a>
<FONT color="green">201</FONT>         * purpose is only to establish initial conditions, it does not really change<a name="line.201"></a>
<FONT color="green">202</FONT>         * time by itself. The top level multistep integrator remains in charge of<a name="line.202"></a>
<FONT color="green">203</FONT>         * handling time propagation and events handling as it will starts its own<a name="line.203"></a>
<FONT color="green">204</FONT>         * computation right from the beginning. In a sense, the starter integrator<a name="line.204"></a>
<FONT color="green">205</FONT>         * can be seen as a dummy one and so it will never trigger any user event nor<a name="line.205"></a>
<FONT color="green">206</FONT>         * call any user step handler.&lt;/p&gt;<a name="line.206"></a>
<FONT color="green">207</FONT>         * @param t0 initial time<a name="line.207"></a>
<FONT color="green">208</FONT>         * @param y0 initial value of the state vector at t0<a name="line.208"></a>
<FONT color="green">209</FONT>         * @param t target time for the integration<a name="line.209"></a>
<FONT color="green">210</FONT>         * (can be set to a value smaller than &lt;code&gt;t0&lt;/code&gt; for backward integration)<a name="line.210"></a>
<FONT color="green">211</FONT>         * @exception DimensionMismatchException if arrays dimension do not match equations settings<a name="line.211"></a>
<FONT color="green">212</FONT>         * @exception NumberIsTooSmallException if integration step is too small<a name="line.212"></a>
<FONT color="green">213</FONT>         * @exception MaxCountExceededException if the number of functions evaluations is exceeded<a name="line.213"></a>
<FONT color="green">214</FONT>         * @exception NoBracketingException if the location of an event cannot be bracketed<a name="line.214"></a>
<FONT color="green">215</FONT>         */<a name="line.215"></a>
<FONT color="green">216</FONT>        protected void start(final double t0, final double[] y0, final double t)<a name="line.216"></a>
<FONT color="green">217</FONT>            throws DimensionMismatchException, NumberIsTooSmallException,<a name="line.217"></a>
<FONT color="green">218</FONT>                   MaxCountExceededException, NoBracketingException {<a name="line.218"></a>
<FONT color="green">219</FONT>    <a name="line.219"></a>
<FONT color="green">220</FONT>            // make sure NO user event nor user step handler is triggered,<a name="line.220"></a>
<FONT color="green">221</FONT>            // this is the task of the top level integrator, not the task<a name="line.221"></a>
<FONT color="green">222</FONT>            // of the starter integrator<a name="line.222"></a>
<FONT color="green">223</FONT>            starter.clearEventHandlers();<a name="line.223"></a>
<FONT color="green">224</FONT>            starter.clearStepHandlers();<a name="line.224"></a>
<FONT color="green">225</FONT>    <a name="line.225"></a>
<FONT color="green">226</FONT>            // set up one specific step handler to extract initial Nordsieck vector<a name="line.226"></a>
<FONT color="green">227</FONT>            starter.addStepHandler(new NordsieckInitializer(nSteps, y0.length));<a name="line.227"></a>
<FONT color="green">228</FONT>    <a name="line.228"></a>
<FONT color="green">229</FONT>            // start integration, expecting a InitializationCompletedMarkerException<a name="line.229"></a>
<FONT color="green">230</FONT>            try {<a name="line.230"></a>
<FONT color="green">231</FONT>                starter.integrate(new CountingDifferentialEquations(y0.length),<a name="line.231"></a>
<FONT color="green">232</FONT>                                  t0, y0, t, new double[y0.length]);<a name="line.232"></a>
<FONT color="green">233</FONT>            } catch (InitializationCompletedMarkerException icme) { // NOPMD<a name="line.233"></a>
<FONT color="green">234</FONT>                // this is the expected nominal interruption of the start integrator<a name="line.234"></a>
<FONT color="green">235</FONT>            }<a name="line.235"></a>
<FONT color="green">236</FONT>    <a name="line.236"></a>
<FONT color="green">237</FONT>            // remove the specific step handler<a name="line.237"></a>
<FONT color="green">238</FONT>            starter.clearStepHandlers();<a name="line.238"></a>
<FONT color="green">239</FONT>    <a name="line.239"></a>
<FONT color="green">240</FONT>        }<a name="line.240"></a>
<FONT color="green">241</FONT>    <a name="line.241"></a>
<FONT color="green">242</FONT>        /** Initialize the high order scaled derivatives at step start.<a name="line.242"></a>
<FONT color="green">243</FONT>         * @param h step size to use for scaling<a name="line.243"></a>
<FONT color="green">244</FONT>         * @param t first steps times<a name="line.244"></a>
<FONT color="green">245</FONT>         * @param y first steps states<a name="line.245"></a>
<FONT color="green">246</FONT>         * @param yDot first steps derivatives<a name="line.246"></a>
<FONT color="green">247</FONT>         * @return Nordieck vector at first step (h&lt;sup&gt;2&lt;/sup&gt;/2 y''&lt;sub&gt;n&lt;/sub&gt;,<a name="line.247"></a>
<FONT color="green">248</FONT>         * h&lt;sup&gt;3&lt;/sup&gt;/6 y'''&lt;sub&gt;n&lt;/sub&gt; ... h&lt;sup&gt;k&lt;/sup&gt;/k! y&lt;sup&gt;(k)&lt;/sup&gt;&lt;sub&gt;n&lt;/sub&gt;)<a name="line.248"></a>
<FONT color="green">249</FONT>         */<a name="line.249"></a>
<FONT color="green">250</FONT>        protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,<a name="line.250"></a>
<FONT color="green">251</FONT>                                                                               final double[][] y,<a name="line.251"></a>
<FONT color="green">252</FONT>                                                                               final double[][] yDot);<a name="line.252"></a>
<FONT color="green">253</FONT>    <a name="line.253"></a>
<FONT color="green">254</FONT>        /** Get the minimal reduction factor for stepsize control.<a name="line.254"></a>
<FONT color="green">255</FONT>         * @return minimal reduction factor<a name="line.255"></a>
<FONT color="green">256</FONT>         */<a name="line.256"></a>
<FONT color="green">257</FONT>        public double getMinReduction() {<a name="line.257"></a>
<FONT color="green">258</FONT>            return minReduction;<a name="line.258"></a>
<FONT color="green">259</FONT>        }<a name="line.259"></a>
<FONT color="green">260</FONT>    <a name="line.260"></a>
<FONT color="green">261</FONT>        /** Set the minimal reduction factor for stepsize control.<a name="line.261"></a>
<FONT color="green">262</FONT>         * @param minReduction minimal reduction factor<a name="line.262"></a>
<FONT color="green">263</FONT>         */<a name="line.263"></a>
<FONT color="green">264</FONT>        public void setMinReduction(final double minReduction) {<a name="line.264"></a>
<FONT color="green">265</FONT>            this.minReduction = minReduction;<a name="line.265"></a>
<FONT color="green">266</FONT>        }<a name="line.266"></a>
<FONT color="green">267</FONT>    <a name="line.267"></a>
<FONT color="green">268</FONT>        /** Get the maximal growth factor for stepsize control.<a name="line.268"></a>
<FONT color="green">269</FONT>         * @return maximal growth factor<a name="line.269"></a>
<FONT color="green">270</FONT>         */<a name="line.270"></a>
<FONT color="green">271</FONT>        public double getMaxGrowth() {<a name="line.271"></a>
<FONT color="green">272</FONT>            return maxGrowth;<a name="line.272"></a>
<FONT color="green">273</FONT>        }<a name="line.273"></a>
<FONT color="green">274</FONT>    <a name="line.274"></a>
<FONT color="green">275</FONT>        /** Set the maximal growth factor for stepsize control.<a name="line.275"></a>
<FONT color="green">276</FONT>         * @param maxGrowth maximal growth factor<a name="line.276"></a>
<FONT color="green">277</FONT>         */<a name="line.277"></a>
<FONT color="green">278</FONT>        public void setMaxGrowth(final double maxGrowth) {<a name="line.278"></a>
<FONT color="green">279</FONT>            this.maxGrowth = maxGrowth;<a name="line.279"></a>
<FONT color="green">280</FONT>        }<a name="line.280"></a>
<FONT color="green">281</FONT>    <a name="line.281"></a>
<FONT color="green">282</FONT>        /** Get the safety factor for stepsize control.<a name="line.282"></a>
<FONT color="green">283</FONT>         * @return safety factor<a name="line.283"></a>
<FONT color="green">284</FONT>         */<a name="line.284"></a>
<FONT color="green">285</FONT>        public double getSafety() {<a name="line.285"></a>
<FONT color="green">286</FONT>          return safety;<a name="line.286"></a>
<FONT color="green">287</FONT>        }<a name="line.287"></a>
<FONT color="green">288</FONT>    <a name="line.288"></a>
<FONT color="green">289</FONT>        /** Set the safety factor for stepsize control.<a name="line.289"></a>
<FONT color="green">290</FONT>         * @param safety safety factor<a name="line.290"></a>
<FONT color="green">291</FONT>         */<a name="line.291"></a>
<FONT color="green">292</FONT>        public void setSafety(final double safety) {<a name="line.292"></a>
<FONT color="green">293</FONT>          this.safety = safety;<a name="line.293"></a>
<FONT color="green">294</FONT>        }<a name="line.294"></a>
<FONT color="green">295</FONT>    <a name="line.295"></a>
<FONT color="green">296</FONT>        /** Compute step grow/shrink factor according to normalized error.<a name="line.296"></a>
<FONT color="green">297</FONT>         * @param error normalized error of the current step<a name="line.297"></a>
<FONT color="green">298</FONT>         * @return grow/shrink factor for next step<a name="line.298"></a>
<FONT color="green">299</FONT>         */<a name="line.299"></a>
<FONT color="green">300</FONT>        protected double computeStepGrowShrinkFactor(final double error) {<a name="line.300"></a>
<FONT color="green">301</FONT>            return FastMath.min(maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp)));<a name="line.301"></a>
<FONT color="green">302</FONT>        }<a name="line.302"></a>
<FONT color="green">303</FONT>    <a name="line.303"></a>
<FONT color="green">304</FONT>        /** Transformer used to convert the first step to Nordsieck representation. */<a name="line.304"></a>
<FONT color="green">305</FONT>        public interface NordsieckTransformer {<a name="line.305"></a>
<FONT color="green">306</FONT>            /** Initialize the high order scaled derivatives at step start.<a name="line.306"></a>
<FONT color="green">307</FONT>             * @param h step size to use for scaling<a name="line.307"></a>
<FONT color="green">308</FONT>             * @param t first steps times<a name="line.308"></a>
<FONT color="green">309</FONT>             * @param y first steps states<a name="line.309"></a>
<FONT color="green">310</FONT>             * @param yDot first steps derivatives<a name="line.310"></a>
<FONT color="green">311</FONT>             * @return Nordieck vector at first step (h&lt;sup&gt;2&lt;/sup&gt;/2 y''&lt;sub&gt;n&lt;/sub&gt;,<a name="line.311"></a>
<FONT color="green">312</FONT>             * h&lt;sup&gt;3&lt;/sup&gt;/6 y'''&lt;sub&gt;n&lt;/sub&gt; ... h&lt;sup&gt;k&lt;/sup&gt;/k! y&lt;sup&gt;(k)&lt;/sup&gt;&lt;sub&gt;n&lt;/sub&gt;)<a name="line.312"></a>
<FONT color="green">313</FONT>             */<a name="line.313"></a>
<FONT color="green">314</FONT>            Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,<a name="line.314"></a>
<FONT color="green">315</FONT>                                                                final double[][] y,<a name="line.315"></a>
<FONT color="green">316</FONT>                                                                final double[][] yDot);<a name="line.316"></a>
<FONT color="green">317</FONT>        }<a name="line.317"></a>
<FONT color="green">318</FONT>    <a name="line.318"></a>
<FONT color="green">319</FONT>        /** Specialized step handler storing the first step. */<a name="line.319"></a>
<FONT color="green">320</FONT>        private class NordsieckInitializer implements StepHandler {<a name="line.320"></a>
<FONT color="green">321</FONT>    <a name="line.321"></a>
<FONT color="green">322</FONT>            /** Steps counter. */<a name="line.322"></a>
<FONT color="green">323</FONT>            private int count;<a name="line.323"></a>
<FONT color="green">324</FONT>    <a name="line.324"></a>
<FONT color="green">325</FONT>            /** First steps times. */<a name="line.325"></a>
<FONT color="green">326</FONT>            private final double[] t;<a name="line.326"></a>
<FONT color="green">327</FONT>    <a name="line.327"></a>
<FONT color="green">328</FONT>            /** First steps states. */<a name="line.328"></a>
<FONT color="green">329</FONT>            private final double[][] y;<a name="line.329"></a>
<FONT color="green">330</FONT>    <a name="line.330"></a>
<FONT color="green">331</FONT>            /** First steps derivatives. */<a name="line.331"></a>
<FONT color="green">332</FONT>            private final double[][] yDot;<a name="line.332"></a>
<FONT color="green">333</FONT>    <a name="line.333"></a>
<FONT color="green">334</FONT>            /** Simple constructor.<a name="line.334"></a>
<FONT color="green">335</FONT>             * @param nSteps number of steps of the multistep method (excluding the one being computed)<a name="line.335"></a>
<FONT color="green">336</FONT>             * @param n problem dimension<a name="line.336"></a>
<FONT color="green">337</FONT>             */<a name="line.337"></a>
<FONT color="green">338</FONT>            public NordsieckInitializer(final int nSteps, final int n) {<a name="line.338"></a>
<FONT color="green">339</FONT>                this.count = 0;<a name="line.339"></a>
<FONT color="green">340</FONT>                this.t     = new double[nSteps];<a name="line.340"></a>
<FONT color="green">341</FONT>                this.y     = new double[nSteps][n];<a name="line.341"></a>
<FONT color="green">342</FONT>                this.yDot  = new double[nSteps][n];<a name="line.342"></a>
<FONT color="green">343</FONT>            }<a name="line.343"></a>
<FONT color="green">344</FONT>    <a name="line.344"></a>
<FONT color="green">345</FONT>            /** {@inheritDoc} */<a name="line.345"></a>
<FONT color="green">346</FONT>            public void handleStep(StepInterpolator interpolator, boolean isLast)<a name="line.346"></a>
<FONT color="green">347</FONT>                throws MaxCountExceededException {<a name="line.347"></a>
<FONT color="green">348</FONT>    <a name="line.348"></a>
<FONT color="green">349</FONT>                final double prev = interpolator.getPreviousTime();<a name="line.349"></a>
<FONT color="green">350</FONT>                final double curr = interpolator.getCurrentTime();<a name="line.350"></a>
<FONT color="green">351</FONT>    <a name="line.351"></a>
<FONT color="green">352</FONT>                if (count == 0) {<a name="line.352"></a>
<FONT color="green">353</FONT>                    // first step, we need to store also the beginning of the step<a name="line.353"></a>
<FONT color="green">354</FONT>                    interpolator.setInterpolatedTime(prev);<a name="line.354"></a>
<FONT color="green">355</FONT>                    t[0] = prev;<a name="line.355"></a>
<FONT color="green">356</FONT>                    System.arraycopy(interpolator.getInterpolatedState(), 0,<a name="line.356"></a>
<FONT color="green">357</FONT>                                     y[0],    0, y[0].length);<a name="line.357"></a>
<FONT color="green">358</FONT>                    System.arraycopy(interpolator.getInterpolatedDerivatives(), 0,<a name="line.358"></a>
<FONT color="green">359</FONT>                                     yDot[0], 0, yDot[0].length);<a name="line.359"></a>
<FONT color="green">360</FONT>                }<a name="line.360"></a>
<FONT color="green">361</FONT>    <a name="line.361"></a>
<FONT color="green">362</FONT>                // store the end of the step<a name="line.362"></a>
<FONT color="green">363</FONT>                ++count;<a name="line.363"></a>
<FONT color="green">364</FONT>                interpolator.setInterpolatedTime(curr);<a name="line.364"></a>
<FONT color="green">365</FONT>                t[count] = curr;<a name="line.365"></a>
<FONT color="green">366</FONT>                System.arraycopy(interpolator.getInterpolatedState(), 0,<a name="line.366"></a>
<FONT color="green">367</FONT>                                 y[count],    0, y[count].length);<a name="line.367"></a>
<FONT color="green">368</FONT>                System.arraycopy(interpolator.getInterpolatedDerivatives(), 0,<a name="line.368"></a>
<FONT color="green">369</FONT>                                 yDot[count], 0, yDot[count].length);<a name="line.369"></a>
<FONT color="green">370</FONT>    <a name="line.370"></a>
<FONT color="green">371</FONT>                if (count == t.length - 1) {<a name="line.371"></a>
<FONT color="green">372</FONT>    <a name="line.372"></a>
<FONT color="green">373</FONT>                    // this was the last step we needed, we can compute the derivatives<a name="line.373"></a>
<FONT color="green">374</FONT>                    stepStart = t[0];<a name="line.374"></a>
<FONT color="green">375</FONT>                    stepSize  = (t[t.length - 1] - t[0]) / (t.length - 1);<a name="line.375"></a>
<FONT color="green">376</FONT>    <a name="line.376"></a>
<FONT color="green">377</FONT>                    // first scaled derivative<a name="line.377"></a>
<FONT color="green">378</FONT>                    scaled = yDot[0].clone();<a name="line.378"></a>
<FONT color="green">379</FONT>                    for (int j = 0; j &lt; scaled.length; ++j) {<a name="line.379"></a>
<FONT color="green">380</FONT>                        scaled[j] *= stepSize;<a name="line.380"></a>
<FONT color="green">381</FONT>                    }<a name="line.381"></a>
<FONT color="green">382</FONT>    <a name="line.382"></a>
<FONT color="green">383</FONT>                    // higher order derivatives<a name="line.383"></a>
<FONT color="green">384</FONT>                    nordsieck = initializeHighOrderDerivatives(stepSize, t, y, yDot);<a name="line.384"></a>
<FONT color="green">385</FONT>    <a name="line.385"></a>
<FONT color="green">386</FONT>                    // stop the integrator now that all needed steps have been handled<a name="line.386"></a>
<FONT color="green">387</FONT>                    throw new InitializationCompletedMarkerException();<a name="line.387"></a>
<FONT color="green">388</FONT>    <a name="line.388"></a>
<FONT color="green">389</FONT>                }<a name="line.389"></a>
<FONT color="green">390</FONT>    <a name="line.390"></a>
<FONT color="green">391</FONT>            }<a name="line.391"></a>
<FONT color="green">392</FONT>    <a name="line.392"></a>
<FONT color="green">393</FONT>            /** {@inheritDoc} */<a name="line.393"></a>
<FONT color="green">394</FONT>            public void init(double t0, double[] y0, double time) {<a name="line.394"></a>
<FONT color="green">395</FONT>                // nothing to do<a name="line.395"></a>
<FONT color="green">396</FONT>            }<a name="line.396"></a>
<FONT color="green">397</FONT>    <a name="line.397"></a>
<FONT color="green">398</FONT>        }<a name="line.398"></a>
<FONT color="green">399</FONT>    <a name="line.399"></a>
<FONT color="green">400</FONT>        /** Marker exception used ONLY to stop the starter integrator after first step. */<a name="line.400"></a>
<FONT color="green">401</FONT>        private static class InitializationCompletedMarkerException<a name="line.401"></a>
<FONT color="green">402</FONT>            extends RuntimeException {<a name="line.402"></a>
<FONT color="green">403</FONT>    <a name="line.403"></a>
<FONT color="green">404</FONT>            /** Serializable version identifier. */<a name="line.404"></a>
<FONT color="green">405</FONT>            private static final long serialVersionUID = -1914085471038046418L;<a name="line.405"></a>
<FONT color="green">406</FONT>    <a name="line.406"></a>
<FONT color="green">407</FONT>            /** Simple constructor. */<a name="line.407"></a>
<FONT color="green">408</FONT>            public InitializationCompletedMarkerException() {<a name="line.408"></a>
<FONT color="green">409</FONT>                super((Throwable) null);<a name="line.409"></a>
<FONT color="green">410</FONT>            }<a name="line.410"></a>
<FONT color="green">411</FONT>    <a name="line.411"></a>
<FONT color="green">412</FONT>        }<a name="line.412"></a>
<FONT color="green">413</FONT>    <a name="line.413"></a>
<FONT color="green">414</FONT>        /** Wrapper for differential equations, ensuring start evaluations are counted. */<a name="line.414"></a>
<FONT color="green">415</FONT>        private class CountingDifferentialEquations implements FirstOrderDifferentialEquations {<a name="line.415"></a>
<FONT color="green">416</FONT>    <a name="line.416"></a>
<FONT color="green">417</FONT>            /** Dimension of the problem. */<a name="line.417"></a>
<FONT color="green">418</FONT>            private final int dimension;<a name="line.418"></a>
<FONT color="green">419</FONT>    <a name="line.419"></a>
<FONT color="green">420</FONT>            /** Simple constructor.<a name="line.420"></a>
<FONT color="green">421</FONT>             * @param dimension dimension of the problem<a name="line.421"></a>
<FONT color="green">422</FONT>             */<a name="line.422"></a>
<FONT color="green">423</FONT>            public CountingDifferentialEquations(final int dimension) {<a name="line.423"></a>
<FONT color="green">424</FONT>                this.dimension = dimension;<a name="line.424"></a>
<FONT color="green">425</FONT>            }<a name="line.425"></a>
<FONT color="green">426</FONT>    <a name="line.426"></a>
<FONT color="green">427</FONT>            /** {@inheritDoc} */<a name="line.427"></a>
<FONT color="green">428</FONT>            public void computeDerivatives(double t, double[] y, double[] dot)<a name="line.428"></a>
<FONT color="green">429</FONT>                throws MaxCountExceededException, DimensionMismatchException {<a name="line.429"></a>
<FONT color="green">430</FONT>                MultistepIntegrator.this.computeDerivatives(t, y, dot);<a name="line.430"></a>
<FONT color="green">431</FONT>            }<a name="line.431"></a>
<FONT color="green">432</FONT>    <a name="line.432"></a>
<FONT color="green">433</FONT>            /** {@inheritDoc} */<a name="line.433"></a>
<FONT color="green">434</FONT>            public int getDimension() {<a name="line.434"></a>
<FONT color="green">435</FONT>                return dimension;<a name="line.435"></a>
<FONT color="green">436</FONT>            }<a name="line.436"></a>
<FONT color="green">437</FONT>    <a name="line.437"></a>
<FONT color="green">438</FONT>        }<a name="line.438"></a>
<FONT color="green">439</FONT>    <a name="line.439"></a>
<FONT color="green">440</FONT>    }<a name="line.440"></a>




























































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